COORDINATES OE A CUBIC CURVE IN SPACE. 
383 
and if in these last determinants we subtract rows 4, 5, 6 from rows 1, 2, 3 respec¬ 
tively ; and then subtract column 3 from column 4, we shall find that the whole 
expression 
= | (5, y, 8 | { | /3', 8, S' | x | y, 8, S' | — | y', 8, S' | x | ft, 8, S' | } 
= — I ft, y, S | (D, D, y, z), 
or more simply 
= i ft y, 8 I p, y, P 
From this result we may at once conclude the following group: 
&= ! ft y, § | (D, y, z) .(17) 
33= | y, «, S j (D, z, x), 
@= | «, ft 8 | P, x > y )• 
Again, proceeding as before we should find 
-2#=2(AD y D,-HD,D. r -GD,D y +FD/) (A, H, . . . suffix t) 
= 
{ a, a', i 
S| + 
I a , a ', 
s 111 S, 8 
', ft 1 x 
S, S', y | 
— 
{ | ft, < 
51 + 
| ft 
S |}| ft 8 
', a | X 
1 S, S', y | 
— 
{ ! y, a ', < 
51 + 
I /> 
S |H ft 8 
ft ft I X 
| S, S', a | 
+ 
{ 1 ft y,; 
51 + 
1 y> ft', 
S 111 ft 8 
', a | X 
! S, 8', a | 
:a, 
s, . . 
ft x 
1 s, S', • 
y X a, a'. 
, S, . . 
y X | 8, 8', 
ft 1 
«1, 
8j, . . 
ft 
«i, a i 
ft, . . 
yi 
a a , 
a/, 
■ 8* • 
ft 
a,, a 3 
', ft, . . 
y% 
a, 
. 
• S, S', 
ft 
a, . 
. 8, 8 
'» y 
• ft, 8,' 
. ft 
a i, • 
• ft, 8 
ft yi 
a 3 , 
• 8*8*' 
i ft 
a 3 , . 
• ft, ft 
ft y^ 
( 
a, 
y', S, . 
• ft 
+ a , 
/S', s, . 
. y } x 
| S, 8', a [ 
«i, 
yft ft, . 
• ft 
«i, 
ft', Si, . 
• yi 
a 3 , 
yft So, . 
• ft 
«2, 
ft', 8/, . 
• y-2 
a, 
• • S, 
S', ft 
a. 
. . 8, 
S', y 
a i, 
• • K 
Sft ft 
“l, 
■ ■ ft, 
8/, 7i 
a 2 , 
• • ft, 
S/, ft 
a 3 , 
• • ft, 
So', yo 
- I 8, 
S', 
VL 
X 
| S, S', y 
1 X 1 
a, 8, ft 
| — I S, S', 
a' | X | 
s, S', ft 1 x 1 
a, S, y 
+ I 8, 
S', 
a 
1 x 
| S, S', y 
i x | 
«, 8, ft + | 8, S', 
« 1 X 1 
8, S', ft' | x | 
«, S, y 
a, 8, ft j (D, z,x) + I y, a, 8 | (D, a, y). 
